Operator Generic Fundamentals Reactor Theory - Reactivity Coefficients

Operator Generic Fundamentals Reactor Theory - Reactivity Coefficients
Ensure students have calculators that are permitted for use on the Generic Fundamentals Examination. K1.01 Define moderator temperature coefficient of reactivity K1.02 Define fuel temperature coefficient of reactivity K1.03 Describe the effect on the magnitude of the temperature coefficient of reactivity from changes in moderator temperature and core age K1.04 Explain resonance absorption K1.05 Explain doppler broadening and self-shielding K1.06 Describe time effects of core age, moderator temperature, and boron concentration on moderator temperature coefficients K1.07 Describe the effects of core age, fuel temperature, and moderator temperature on fuel temperature (doppler) coefficient K1.08 Describe the components of power coefficient K1.09 Compare boron reactivity worth vs. boron concentration K1.10 Compare boron reactivity worth vs. moderator temperature K1.11 Explain the change in reactivity addition rate due to boration/dilution over core life K1.12 Explain differences between reactivity coefficients and reactivity defects K1.13 Explain and describe the effect of power defect and doppler defect on reactivity Operator Generic Fundamentals Reactor Theory - Reactivity Coefficients

Reactivity Coefficients Introduction
This module discusses reactivity, reactivity coefficients, and defects applied to reactor operations. Core reactivity changes with changes in: Fuel loading and burnup Moderator and fuel temperature Boron concentration Power level Fission product poisons These changes are quantified by coefficients and defects and have dynamic effects with each other (some of which conflict with others). INTRO

Terminal Learning Objectives
At the completion of this training session, the trainee will demonstrate mastery of this topic by passing a written exam with a grade of ≥ 80 percent on the following TLOs: Describe moderator, void and pressure reactivity coefficients and how they are affected by changing reactor conditions. Describe the fuel temperature and power reactivity coefficients and describe how they are affected by changing reactor conditions. Describe boron worth and how boron is utilized to control the reactor from beginning to end of core life. TLO

Reactivity and Moderator Temperature Coefficients
TLO 1 – Describe moderator, void, and pressure reactivity coefficients and how they are affected by changing reactor conditions. 1.1 Explain differences between reactivity coefficients and reactivity defects and how they are used to balance reactivity parameters. 1.2 Describe the moderator temperature coefficient of reactivity. 1.3 Describe how the magnitude of the moderator temperature coefficient varies with changes in the following parameters: Overmoderation and undermoderation of moderator-to-fuel ratio Moderator temperature Boron concentration Control Rod position Core age 1.4 Describe the void and pressure coefficients of reactivity. 3.a) – really a combination of boron and temperature, but the concept of overmoderated and undermoderated is tested a lot by the NRC. 4) – void and pressure coefficients are inconsequential in PWRs. Pressure coefficient not tested, void coefficient tested minimally. TLO 1

Reactivity Coefficients
ELO 1.1 – Explain differences between reactivity coefficients and reactivity defects and how they are used to balance reactivity parameters. Coefficient – UNIT change Defect – TOTAL change Several factors affect reactivity (six-factor formula) Specifically affects on: Resonance Escape Probability (p) Thermal Utilization Factor (f) Coefficient changes only consider ONE parameter change at a time Related KA – K1.12 Explain differences between reactivity coefficients and reactivity defects Reactivity can be affected by many factors, such as fuel depletion, temperature, pressure, or fission product poisons Factors affecting reactivity and how they control or predict reactor behavior are discussed STRESS the importance on the last bullet. You can only look at one change at a time. For example raising moderator temperature under two cases. Case 1 Case 2 °F °F same boron same boron or, 1000 ppm ppm °F °F (same temperatures) You cannot evaluate raising moderator temperature from °F at BOL to °F at EOL (because there are different temps (density) and different boron concentration) ELO 1.1

Change in reactivity (Δρ) due to unit change in some associated parameter (x) 𝛼 𝑥 = ∆𝜌 ∆𝑥 Where: αx = reactivity coefficient for plant parameter x Δρ = change in reactivity (Δk/k) Δx = change in some plant parameter Units for MTC are pcm/ °F or ∆k/k/°F Used to quantify the effect from a UNIT variation in a reactor parameter Per °F, per ppm boron, etc. Example, a moderator temperature increase causes a decrease in the reactivity of the core Amount of reactivity change per degree of moderator temperature is the moderator temperature coefficient, MTC Reactivity coefficients are generally symbolized by αx, where x represents some variable reactor parameter that affects reactivity. Important for reactor operator to know how change in any of plant parameter will affect reactor power. Allows operator to predict reactor response during plant evolutions and transients that involve changes in plant operating parameters ELO 1.1

Reactivity Defects Reactivity defect - total amount of reactivity added, positive or negative, due to changing nuclear reactor parameter by a given amount 𝜌 𝑥 = ∆𝑥 𝛼 𝑥 𝜌 𝑥 = ∆𝑥 ∆𝜌 ∆𝑥 Where: ρx = reactivity defect (Δk/k) x = specific parameter (fuel temp, moderator temp, etc.) Δx = change in parameter x x = parameter x reactivity coefficient ELO 1.1

Reactivity Defects Example The moderator temperature coefficient for a reactor is -8.2 pcm/ °F. Calculate the reactivity defect that results from a temperature decrease of 5°F. 𝜌 𝑥 = ∆𝑥 𝛼 𝑥 Solution ∆𝜌= 𝛼 𝑥 ∆ 𝑡 ∆𝜌= −8.2 𝑝𝑐𝑚 ℉ −5 ℉ ∆𝜌=41 𝑝𝑐𝑚 NOTE: You MUST use correct +/- signs when answering these types of questions! For example, a “decrease” in temperature is a (-) negative change in temperature. ELO 1.1

Knowledge Check Moderator temperature coefficient describes the change in reactivity per degree change in _______________. fuel temperature fuel clad temperature reactor vessel temperature reactor coolant temperature Correct answer is D. Correct answer is D. NRC Bank Question – P133 Keep in mind that a coefficient is a UNIT change and a defect is a TOTAL change. ELO 1.1

Moderator Temperature Coefficient
ELO 1.2 – Describe the moderator temperature coefficient of reactivity. Moderator temperature coefficient (MTC) of reactivity is defined as change in reactivity per degree change in moderator temperature Can be a positive or negative change Depends on magnitude in changes of “P” and “f” Related KA K1.01 Define moderator temperature coefficient of reactivity This is probably the most important concept in todays chapter. Understanding the following competing changes will help answer half of today’s bank questions: PUT THIS ON THE BOARD: (P) - Temp increase, density decrease, neutrons travel farther, more likely to be captured, resonance escape probability decreases, negative reactivity added. (f) – Temp increase, density decrease, less competition from moderator, thermal utilization factor Increases, positive reactivity added. ELO 1.2

Change in reactivity per unit change in temperature (°F) of moderator 𝛼 𝑚 = ∆𝜌 ∆𝑇 𝑚𝑜𝑑 = 𝜌 𝑓𝑖𝑛𝑎𝑙 − 𝜌 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑇 𝑚𝑜𝑑 𝑓𝑖𝑛𝑎𝑙 − 𝑇 𝑚𝑜𝑑 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 Where: αm = MTC (Δk/k/°F) or pcm/°F Δρ = change in reactivity with change in mod temp (Δk/k) ΔTmod = change in mod temp (F) Magnitude and sign (+ or -) of MTC is primarily a function of the moderator-to-fuel ratio Commercial PWRs are designed with an undermoderated moderator-to-fuel ratio that provides a negative moderator temperature coefficient Except sometimes early in core life or cold shutdown conditions Overmoderated reactor will have positive MTC The symbols m and T are both used to represent moderator temperature coefficient. In this text, the symbol αm will be used. ELO 1.2

12 Moderator Temperature Coefficient Example A reactor is operating at 480 °F with an effective multiplication factor of (keff = 1.0). The moderator temperature is increased to 490°F and keff decreases to What is the value of the moderator temperature coefficient? Example A reactor is operating at 560°F with an effective multiplication factor of (keff = 1.0). The moderator temperature increases to 570°F and keff decreases to What is the value of the moderator temperature coefficient? Solution First, convert keff values to reactivity. 𝜌= 𝑘 𝑒𝑓𝑓 −1 𝑘 𝑒𝑓𝑓 𝜌 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 = 1−1 1 =0 𝜌 𝑓𝑖𝑛𝑎𝑙 = 0.999− =−1.001× 10 −3 ELO 2.2 ELO 1.2

13 Moderator Temperature Coefficient Example A reactor is operating at 480 °F with an effective multiplication factor of (keff = 1.0). The moderator temperature is increased to 490°F and keff decreases to What is the value of the moderator temperature coefficient? Solution Continued Then, calculate the value of MTC: 𝛼 𝑚 = ∆𝜌 ∆𝑇 𝑚𝑜𝑑 = 𝜌 𝑓𝑖𝑛𝑎𝑙 − 𝜌 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑇 𝑚𝑜𝑑 𝑓𝑖𝑛𝑎𝑙 − 𝑇 𝑚𝑜𝑑 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝛼 𝑚 = −1.001 × 10 −3 −(0) 570 ℉−560 ℉ 𝛼 𝑚 = −1.001× 10 −3 ∆𝑘\𝑘 10 ℉ 𝛼 𝑚 =−1.001× 10 −4 ∆𝑘/𝑘 ℉ =−10 𝑝𝑐𝑚 ℉ ELO 2.2 ELO 1.2

Value of Moderator Temperature Coefficient
A good approximation of MTC is -110-4Δk/k/°F ≈ -10 pcm/oF (BOL, 100% power, equilibrium Xe) ≈ -40 pcm/oF (EOL, 100% power, equilibrium Xe) Affected by boron concentration/time in core life BOL – more boron “pushed” out of core on temp increase “f” increases more EOL – less boron “pushed” out of core on temp increase “f” increases less Net effect MTC gets more negative over core life Negative MTC more desirable because of power level regulating effect Turns power when power increases, causes moderator temperature to increase Moderator temperature increase adds negative reactivity, causing reactor power to stop increasing Power levels off at new value depending on value of MTC and initial reactivity insertion This concept will be explained further in next objective. ELO 1.2

Moderator Temperature Coefficient Calculation
Knowledge Check A reactor is operating at 560°F with keff = The reactor operator borates the reactor an equivalent of 200 pcm (negative reactivity). RCS temperature responds by dropping 10 degrees. Assuming no other reactivity effects, what is MTC? -5 x 10-4 Δk/k/°F -20 x 10-4 Δk/k/°F -10 x 10-4 Δk/k/°F -2 x 10-4 Δk/k/°F Correct answer is D. Correct Answer is D Divide the 200 pcm by 10 to obtain the reactivity change from the RCS/moderator temperature response to the negative reactivity added by boron. This = 20 pcm/°F or 2 x 10-4 Δk/k/°F. Since the drop in temperature is adding positive reactivity this means that the MTC is a negative value. ELO 1.2

Moderator Temperature Coefficient Variations
ELO 1.3 – Describe how the magnitude of the moderator temperature coefficient varies with changes in the following parameters: overmoderation and undermoderation of the moderator-to-fuel ratio, moderator temperature, boron concentration, control rod position, and core age. MTC values not constant throughout core life Moderator-to-fuel ratio can make MTC positive or negative and affect its magnitude “moderator-fuel-ratio” is mostly about DENSITY Related KAs K1.03, Describe the effect on the magnitude of the temperature coefficient of reactivity from changes in moderator temperature and core age ; K1.06 Describe time effects of core age, moderator temperature, and boron concentration on moderator temperature coefficients ELO 1.3

Moderator to Fuel Ratio Effects on MTC
Changes in moderator-to-fuel ratio (density) affect: Thermal utilization factor (f) Resonance escape probability (P) keff and reactivity, or more precisely the MTC Moderator could be: Undermoderated High temp, low boron Overmoderated Low temp, high boron Reactor designer adjusts the amount of moderator and fuel in the core (Nm/Nu ratio) to an optimum value to ensure a negative MTC throughout core life Moderator temperature, affecting moderator density, causes moderator-to-fuel ratio to change The moderator-to-fuel ratio (Nm/Nu), is very important in the discussion of moderators. Once again, keep in mind that the term “moderation-to-fuel-ratio” can be understood as “density”. ELO 1.3

Undermoderation, Temperature Increasing
As moderator temperature increases Density decreases Moderator-to-fuel ratio decreases Neutrons travel further, greater chance of resonance capture Resonance escape probability decreases Also, fewer moderator atoms means less competition from moderator Thermal utilization increases Total effect - resonance escape probability (P) decreases greater than the increase for thermal utilization (f) keff decreases PWRs are designed to be undermoderated Leads to a negative temperature coefficient, inherent safety feature See next slide for graphics ELO 1.3

Undermoderation, Temperature Increasing
Moderator temperature increases, density decreases, moderator-to-fuel ratio decreases Resonance escape probability decreases and thermal utilization increases “P” dominant Total effect - Resonance escape probability decrease is greater than the increase for thermal utilization, keff decreases Show how the density change affects the moderator to fuel ratio and the changes to the resonance escape probability and the thermal utilization factor. Temperature Increase Figure: keff versus Moderator-to -Fuel Ratio ELO 1.3

Overmoderation High moderator-to-fuel ratio = overmoderated condition
On a temperature increase, keff increases Decrease in resonance escape probability is smaller Increase to thermal utilization factor is larger Referred to as a +MTC Low power after refueling outage High boron, low temperature Cold shutdown conditions Less of a concern, however Tech Specs determine if allowed Basically +MTC equates to HIGH Boron and LOW temperature. At low powers after a refueling outage where you have high boron concentration (overcome Kexcess). As you continue to go up in power (higher temp) and lower boron concentration, MTC eventually becomes negative and contin ues to get more negative. Some plants allow for a slightly positive MTC, some plants do not. See next slide for graphic ELO 1.3

Overmoderation, Temperature Increasing
“f” dominant Total effect - resonance escape probability decrease is less than increase for thermal utilization keff increases Resonance escape probability decreases and thermal utilization increases Moderator temperature increases, density decreases, moderator-to-fuel ratio decreases Show how the density change affects the moderator to fuel ratio and the changes to the resonance escape probability and the thermal utilization factor. Keep in mind that the change in slope on this side is not as dominant as it is when undermoderated. Tech Specs might allow for a slightly positive MTC depending on how much boron is required to overcome Kexcess. A +MTC means that as you withdraw rods to raise power, Temperature Increase Figure: keff versus Moderator-to -Fuel Ratio ELO 1.3

Limitations on MTC Positive MTC
Limited to ensure reactor is controllable Negative MTC Limited to offset positive reactivity from steam line break accident More negative reactivity when temp increases BUT… More positive reactivity when temp decreases Two points of clarification about the lower absolute value of MTC at lower moderator temperature: Too Positive – reactor may be uncontrollable. NOTE: Even with +MTC, fuel temperature ALWAYS adds negative reactivity so the net effect might still be negative reactivity added Explained in later slides Too Negative – possible restart accident from main steam line break. Cooldown would add more positive reactivity. Safety systems only designed to remove “decay heat”, not “fission heat”. REACTOR MUST REMAIN SUBCRITICAL! MTC directed by Tech Specs Might range from +1 to -35 pcm/degree BOL to EOL ELO 1.3

Moderator Temperature Effects on MTC
Density change greater at higher temperatures Neutrons travel farther if change is 558°F-559°F, versus 121°F-122°F Mention that the greater the density change also the greater the effect to thermal and fast non-leakage. Although not a large factor in MTC the non-leakage terms add to the negative side of MTC. Figure: Water Density Change versus Moderator Temperature ELO 1.3

Moderator Temperature Effects on MTC
When temperature raised at high temps: Bigger decrease in density, bigger decrease in resonance escape factor (large negative reactivity) Minor change in thermal utilization factor Atoms/cm3 versus change in density (slight positive reactivity) Net effect - negative reactivity added MTC is MORE negative at high RCS temps Or high power Recall previous graph As temp increases, move to left on graph Bigger decrease in resonance escape probability (P) Graph of “P” essentially graph of density Second “dash” explanation: At low temps when density it high there are more atoms/cm3 of boron in the core. However, when temp is raised one degree at low temps there is a small change in density, pushing “x” amount of boron out of the core. At high temps when density is low there are fewer atoms/cm3 of boron in the core. However, when temp is raised one degree at high temps there is a larger change in density, pushing “x” amount of boron out of the core. For all practical purposes the “x” amount of boron pushed out of the core is essentially the same, therefore, the bigger impact on reactivity is the change in “P”. Mention that the greater the density change also the greater the effect to thermal and fast non-leakage. Although not a large factor in MTC the non-leakage terms add to the negative side of MTC. ELO 1.3

Boron Concentration [B] Effects on MTC
Boron has a high absorption cross-section for thermal neutrons Competition RCS temperature increase “pushes” boron out of the core Lower density High [B] – more boron “pushed” out of core Bigger increase in “f” Low [B] – less boron “pushed” out of core Smaller increase in “f” For a given decrease in “P”: More negativity reactivity added at low [B] (or EOL) Summary – MTC more negative at low boron concentrations The discussion so far has considered the moderator to be pure water. This makes the moderator-to-fuel ratio effect on MTC easier to explain. However, the moderator is NOT pure water. Ensure class understands why and how boron is used in a commercial PWR Commercial PWRs use soluble boron, referred to as boric acid, added to the moderator to provide a variable reactivity poison for: Control of kexcess Maintaining Tavg in the program band during power changes Compensation for fission product poisons Reactivity adjustment to “trim” the control rods fully withdrawn ELO 1.3

Boron Concentration Effects on MTC
Case 1 [B] = 1000 ppm Raise temp from 500 – 501oF Given decrease in “P” Bigger increase in “f” Smaller “net negative reactivity added Case 2 [B] = 100 ppm Raise temp from 500 – 501oF Same decrease in “P” Smaller increase in “f” Greater “net negative reactivity added Consider the next figure that illustrates the response of the thermal utilization factor (f) on moderator/coolant boron concentration. MTC more negative at low boron concentrations ELO 1.3

Slope at 2000 ppm > slope at “no boron” Note that this graph has been “exaggerated” to visualize the concept of how the slope is greater at 2000 ppm versus 0 ppm. Recall from previous graph on Overmoderation/Undermoderation, the slope of “f” doesn’t change that much from low temps to high temps. Figure: Boron Effect on the Thermal Utilization Factor ELO 1.3

“P” dominant “f” dominant Increasing [B] More +MTC Decreasing [B] More -MTC Even though this graph is based on a change in moderator-to-fuel ratio for a given boron concentration, you can sort of visualize it at a constant temperature with changing boron concentrations. As you raise boron concentration you move to the right (more overmoderated, less negative MTC, less “P” dominant, etc.) As you lower boron concentration you move to the left (more undermoderated, more negative MTC, more “P” dominant, etc.) Silly concepts, but this will help you answer questions: More undermoderated means less overmoderated More negative reactivity means less positive reactivity ELO 1.3

Knowledge Check How does the addition of boric acid to the reactor coolant affect the moderator temperature coefficient (MTC) in an undermoderated reactor? The initially negative MTC becomes more negative. The initially negative MTC becomes less negative. The initially positive MTC becomes more positive. The initially positive MTC becomes less positive. Correct answer is B. Correct answer is B. NRC Bank Question – P3151 Analysis An undermoderated core will initially have a negative MTC because the decrease in resonance escape probability dominates the increase in thermal utilization factor for an increase in average coolant temperature. The moderator temperature coefficient will be less negative when Boron concentration rises. A higher Boron concentration results in more competition with the fuel, therefore an increase in average coolant temperature (less dense) pushes more Boron out of the core, tending to make the MTC less negative. Therefore, the initially negative MTC becomes less negative. Choice “B” is correct. ELO 1.3

Knowledge Check As the reactor coolant boron concentration increases, the moderator temperature coefficient becomes less negative. This is because a 1°F increase in reactor coolant temperature at higher boron concentrations results in a larger increase in the _______________. fast fission factor thermal utilization factor total nonleakage probability resonance escape probability Correct answer is B. Correct answer is B. NRC Bank Question – P50 Analysis (see equation ON CLICK above): An increase in moderator temperature results in a decrease in water density, resulting in an increase in slowing down and diffusion lengths. Because neutrons travel further between collisions, the probability that a neutron may reach the fuel at a resonant energy increases; the resonance escape probability (p) decreases. Conversely, a decrease in density also results in less Hydrogen, Oxygen, and Boron atoms to compete with the fuel. This causes an increase in the thermal utilization factor. For high temperatures, this effect is small and the net effect is that keff lowers for a rise in average coolant temperature. Boron concentration must also be taken into consideration since boron is a neutron absorber. An increase in average coolant temperature, with a high boron concentration (over-moderated, usually at low powers at BOL), will result in the thermal utilization factor increasing more than the resonance escape probability lowers; therefore, a rise in average coolant temperature inserts positive reactivity. Boric acid concentration affects the thermal utilization factor (f) most significantly. If RCS temperature rises (less dense), more boron is pushed out of the core. And the higher the boron concentration, for a given density decrease, more boron is pushed out of the core resulting in “f” increasing more. ELO 1.3

Control Rod Position Effects on MTC
Rods act as leakage boundaries When RCS temperature raised (rods partially inserted) Density decreases, neutron migration length increases More likely for a thermal/epithermal neutron to be captured by a control rod Thermal Non-Leakage probability decreases Negative reactivity is added Therefore, MTC becomes MORE negative with rods inserted See P2150 NRC Bank question for a testing example of this concept. ELO 1.3

Control Rod Position Effects on MTC
If rods inserted makes MTC more negative, rods withdrawn makes MTC more positive Chernobyl Event By withdrawing too many rods, MTC became too positive (+) effect of MTC and void coefficient overcame (-) effect of FTC Power Coefficient became positive 10 CFR 50, General Design Criteria 11 The reactor core and associated coolant systems shall be designed so that in the power operating range the net effect of the prompt inherent nuclear feedback characteristics tends to compensate for a rapid increase in reactivity. Basically, Power Coefficient cannot be positive at power ELO 1.3

Core Age Effects on MTC Similar concept to boron concentration changes: From BOL to EOL boron concentration decreases For a given temperature increase at EOL: Less boron is “pushed” out of the core More negative reactivity is added based on given change in “P” In summary: MTC is most negative at EOL MTC is least negative (or possibly positive) at BOL Commercial PWRs limited on how negative the MTC can become Restriction required because of main steam line break accident During a steam line break accident, the reactor coolant system will undergo a rapid cooldown because steam system begins to act like an infinite heat sink Rapid cooldown will result in large positive reactivity insertion to reactor core from the MTC Some plant accident analyses demonstrate that the reactor could actually become supercritical with all control rods fully inserted An example of such a limit on the MTC is a value such as -44 pcm/F (-4.4  10-4 k/k/F) ELO 1.3

Core Age Effects on MTC Knowledge Check
When compared to the beginning of a fuel cycle, the moderator temperature coefficient at 100 percent power near the end of a fuel cycle is... more negative, because fewer boron-10 nuclei are removed from the core for a given moderator temperature increase less negative, because more boron-10 nuclei are removed from the core for a given moderator temperature increase more negative, because a smaller fraction of the neutron flux will leak out of the core following a given moderator temperature increase less negative, because a larger fraction of the neutron flux will leak out of the core following a given moderator temperature increase Correct answer is A. Correct answer is A. NRC Bank Question – P6126 Analysis: First off, we can eliminate “B” and “D” because MTC is more negative at EOL. A, CORRECT (2/2). The moderator temperature will be most negative at the end of core life with a high RCS temperature. This is because at high temperatures, a rise in temperature will cause a larger decrease in resonance escape probability (p) than it does an increase in thermal utilization factor (f) – see above graph. A low Boron concentration at EOL results in less competition with the fuel, therefore an increase in average coolant temperature (less dense) pushes less Boron out of the core at EOL. At EOL, the change in thermal utilization factor (f) is much smaller than the change in the resonance escape probability (p).. B. WRONG (0/2). The MTC is most negative at end of life (low Boron concentrations) where a rise in average coolant temperature pushes less Boron out of the core. C. WRONG (1/2). Even though MTC is more negative at EOL because of the lower boron concentration at EOL, there is a higher flux at EOL resulting in MORE leakage. Also, the flux moves outwards over core life Resulting in more leakage. However, this is a minimal change. D. WRONG (0/2). The MTC is most negative at end of life (low Boron concentrations) where a rise in average ELO 1.3

Void and Pressure Coefficients
ELO 1.4 – Describe the void and pressure coefficients of reactivity. Void and pressure defects are minor factors Minimal voiding in PWRs at full power RCS essentially incompressible Void coefficient (VC) similar to MTC More voiding as temperature increases Adds negative reactivity Pressure coefficient function of density If pressure increases, density increases, pressure coefficient increases Not related to any specific KA. Void and pressure coefficients play a very small role in reactivity balances for a commercial PWR Rule of thumb for pressure is 100 psi is equal to 1°F temperature change At full power, voids may be about 0.5 percent of total moderator volume Changes in pressure and voiding large enough to make significant reactivity changes in normal operating bands do not occur ELO 1.4

Pressure Coefficient As pressure increases, density increases, increasing the moderator- to-fuel ratio In an undermoderated core, increase in moderator-to-fuel ratio results in positive reactivity addition Pressure coefficient is a positive reactivity coefficient Since RCS maintained at or near 2250 psia, and water incompressible: This coefficient is neglected Pressure coefficient implies that for a given pressure change a certain amount of water density change occurs, which like the moderator temperature effect to density, causes a change in reactivity. Water is incompressible so the Pressure coefficient is very small compared to MTC. ELO 1.4

Void Coefficient Small amount of voiding occurs at high power
Bubbles near top of some fuel assembly do not collapse Voiding results in negative reactivity Neutrons travel farther (more leakage and capture) Voiding at full power Approximately 0.5% Void Coefficient at BOL: ≈ -100pcm/% void Void Coefficient at EOL: ≈ -400pcm/% void Void Defect between -50 and -200 pcm (BOL to EOL) Many plants ignore pressure coefficient and incorporate void coefficient into the moderator temperature coefficient ELO 1.4

Fuel Temperature and Power Coefficients
TLO 2 – Describe the fuel temperature and power reactivity coefficients and how they are affected by changing reactor conditions. 2.1 Describe the fuel temperature coefficient of reactivity. 2.2 Explain resonance absorption, Doppler broadening, and self shielding. 2.3 Describe how the magnitude of the fuel temperature coefficient varies with changes in the following parameters: Moderator temperature Fuel temperature Core age 2.4 Describe the components of the power coefficient of reactivity and the magnitude of their overall effect over core life. 2.5 Explain how the power defect affects the reactivity balance on reactor power operations. All reactivity coefficients and defects are important to understand for safe reactor operations MTC has been mentioned as providing an inherent safety feature for PWRs, but fuel temperature coefficient (Doppler) is just as much an inherent safety feature in that it adds negative reactivity on a power/temperature increase but additionally is faster acting Keep in mind that FTC (like MTC) is the change in reactivity per degree change in fuel temperature. When looking at the factors that effect FTC you can only make ONE change at a time. TLO 2

Fuel Temperature Coefficient
ELO 2.1 – Describe the fuel temperature coefficient of reactivity. FTC - change in reactivity per degree change in fuel temperature Major capture of neutrons while slowing down are Resonant peaks of U-238 and Pu-240 Fuel temperature changes affect the peaks FTC also called Doppler Coefficient Related KA K1.02 Define fuel temperature coefficient of reactivity Reactor power increase causes an immediate increase in fuel temperature to heat the moderator. Recall about 80% of heat from fission is from the kinetic energy of the fission fragments. With NO cladding damage, fission fragments stay in the fuel assembly and heat up fuel as they slow down. In the event of a large positive reactivity insertion, because of the delay in the moderator temperature change, MTC cannot turn reactor power rise for several seconds, whereas FTC starts adding negative reactivity immediately. This reactivity coefficient is very important in commercial PWRs, because unlike the NAVY (high U-235 enrichment and low U-238), commercial PWRs only have about 5% U-235 and 95% U Therefore a lot of capture of neutrons are caused by these resonant peaks on the EVEN numbered fuel isotopes. ELO 2.1

FTC is change in reactivity per unit change in fuel temperature 𝛼 𝐷 = ∆ 𝜌 ∆𝑇 𝑓𝑢𝑒𝑙 = 𝜌 𝑓𝑖𝑛𝑎𝑙 − 𝜌 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑇 𝑓𝑢𝑒𝑙 𝑓𝑖𝑛𝑎𝑙 − 𝑇 𝑓𝑢𝑒𝑙 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 Where: αD = Doppler coefficient (FTC) (Δk/k/°F) Δρ = change in reactivity associated with change in fuel temperature (Δk/k) ΔTfuel = change in fuel temperature (°F) Magnitude of Doppler coefficient in PWRs is about -1 x 10-5 Δk/k/°F or -1 pcm/°F In low enrichment reactor fuel (commercial reactors), most of the uranium found in fuel rods is uranium-238 (plutonium-240 builds in over core life). The importance of the Doppler coefficient is its immediate fuel temperature increase on an increase in power Uranium oxide (fuel pellets) poor conductor of heat and cylindrical fuel rods have a small heat transfer surface per unit volume This means a “relatively” long time period for heat generated at any instant to be transferred to moderator/coolant May be a couple of seconds “relatively” long compared to instantaneous rise in fuel temperature from fission fragment energy MTC is subject to this time delay, and therefore has a delayed effect in countering the positive reactivity ELO 2.1

A fuel temperature increase causes higher vibrational frequency of fuel atoms This increases neutron absorption by U-238 and Pu-240 More neutrons look like resonant energy neutrons In summary: As fuel temperature increases more resonant capture negative reactivity added FTC is ALWAYS negative Unlike MTC What matters to the Fuel Temperature Coefficient is the effective fuel temperature which is less than the average fuel temperature and much less than the peak centerline fuel temperature. Since the self-shielding of the fuel pellet at the resonance energy is so large, most of the resonance absorptions occur on the outer edge of the fuel pellet where the temperatures are much lower than the peak centerline temperature. The effective fuel temperature is a WEIGHTED average fuel temperature that is weighted more by the lower outer edge temperatures. ELO 2.1

Fuel Temperature Defect
Change in fuel temperature from HZP to HFP might average between 900oF to 1500oF (or higher in hotter channels) Depends on fuel channel locations in core Defect = Coefficient x DTfuel Defect = -1pcm/oF x 900oF = -900 pcm These fuel temperatures vary with the different reactor vendors. Although coefficient is small, defect can be a very high value as reactor power level changes from 0 to 100 percent Average fuel temperature at 100 percent reactor power is about 2,200°F; however, peak fuel temperature in some fuel rods could be greater than 3,000°F Magnitude of change in reactivity due to fuel temperature changes is large ELO 2.1

Fuel Temperature Defect
An example of FTD versus reactor core power: This slide can be a little confusing if not explained correctly. Basically, when power Is increased fuel temperature increases. However, it does not increase linearly! Also the amount of reactivity added per degree of fuel temperature change is not linear. However, since the fuel temperature increases at an increasing rate as power is increased and the FTC increases at a decreasing rate, this graph could be representative of the reactivity added by fuel temperature as power is increased. Figure: Doppler Defect vs. Rated Reactor Core Power ELO 2.1

Doppler Broadening and Self Shielding
ELO 2.2 – Explain resonance absorption, doppler broadening, and self shielding. Resonance Absorption U-238 and Pu-240 have high microscopic cross sections for capture U-238 is highest macroscopic cross section (higher “N”) Doppler Broadening Change in resonance peaks due to changes in fuel temp Self-Shielding Relationship between macroscopic cross section and mean free path of neutrons Related KA – K1.04 Explain resonance absorption ; K1.05 Explain doppler broadening and self-shielding. 2.3* 2.4* ELO 2.2

Resonance Absorption U-238/Pu-240 capture neutrons in Intermediate energy range Factor - Resonant Escape Probability Neutrons in this energy range referred to as: Resonant High microscopic cross section for capture (5500 Barns at 21 ev) Off-Resonant Low microscopic cross section for capture ( Barns at 22 or 20 ev) ELO 2.2

Doppler Effect Doppler effect originally associated with physics of sound and light Sound velocity about 730 mph in air Race car moving at 140 mph sounds like 870 mph Coming towards you Race car moving at 140 mph sounds like 590 mph Going away from you Summary: Neutron needs to “look” like a resonant energy neutron for better chance of capture 6.7 ev and 21 ev, for example The probability of resonant neutron absorption as a function of fuel temperature can be explained in terms of the Doppler effect The sound frequency phenomena is explained from a source of sound waves moving toward the listener. Each successive sound wave peak is emitted from a position closer to the listener than the previous sound wave. Therefore, each sound wave takes slightly less time to reach the listener than the previous one, and the time between successive sound wave peaks is also deceasing. This is the increase in sound frequency. The opposite is true if moving away. ELO 2.2

Figure: Resonance Escape Probability Change With Fuel Temperature
Doppler Broadening As fuel temperature increases, resonance capture peaks effectively broaden from increased vibration of nuclei Figure: Resonance Escape Probability Change With Fuel Temperature ELO 2.2

Doppler Broadening Broadening of resonance capture (absorption) peaks occurs as fuel temperature increases, making resonance capture more likely Resonance escape probability decreases Keff decreases Negative reactivity added Recall that the two nuclides of significance here with large resonant peaks are uranium-238 and plutonium-240. In nuclear reactor fuel, this Doppler effect can be used to explain probability of resonant absorption as a function of fuel’s temperature Assume a stationary nuclei will absorb only neutrons of a specific energy Eo If nucleus is moving away from neutron, velocity (and energy) of neutron must be greater than Eo to undergo resonance absorption Likewise, if nucleus is moving toward neutron, neutron needs less energy than Eo to be absorbed Raising nuclei temperature increases rapid vibration within their lattice structures, broadening energy range of neutrons for resonance capture (Doppler broadening) Figure: Doppler Broadening ELO 2.2

Doppler Broadening Figure shows various peaks of U-238
Peaks based on low temp (68oF) Fuel temps much higher 2000oF oF At elevated fuel temperatures: Peaks lower, but broaden Resonant capture relatively constant Off-Resonant capture increases This figure shows the U-238 resonance capture cross-sections as a function of neutron energy for two different fuel temperature conditions; room temperature vs. reactor operating conditions. Resonance capture cross-section for uranium-238 is affected by the relative motion between incident neutron and target nucleus As temperature increases, average kinetic energy of U-238 nucleus increases With increasing temperature, cross-section peak decreases, but energy spectrum broadens Overall, likelihood of a neutron capture increases KE or vibration of nucleus has a direct impact on its magnitude of capture cross-section Figure: Uranium-238 Cross-Section for Absorption Curve ELO 2.2

Doppler Broadening Figure a: At low temp, 21 ev captured 21 + 0 = 21
Figure b: At high temp, 20 ev captured = 21 Figure c: At high temp, 22 ev captured 22 – 1 = 21 In ALL cases, neutron looked like a 21 ev neutron to the U- 238 nucleus Resonant energy cross section actually decreases at higher temps To demonstrate this doppler effect with different neutron and nucleus energies, consider the three neutron reactions depicted in the following figures (a. b. & c.). Figure a: Incident neutron of 21 eV of KE hits a target nucleus at room temperature (roughly eV) Microscopic absorption cross-section U-238 at 21 eV is 5,500 barns - neutron likely to be absorbed Figure b: A 20 eV neutron striking a nucleus vibrating toward it with KE 1 eV Relative energy between incident neutron and target U-238 nucleus is 21 eV As with figure a, the effective absorption cross-section is about 5,500 barns and neutron likely to be absorbed The Student Guide shows another visual aid to help explain Doppler Broadening, but basically a neutron needs to “look” like a resonant energy neutron. Figure: Doppler Effect in Uranium-238 Resonance Capture ELO 2.2

Doppler Broadening Summary
Increased fuel temp results in modified capture cross-section curve: Resonant energy cross section decreases Off-Resonant energy cross section increases More resonant capture, negative reactivity added However, area under both curves is theoretically same Based on change to cross section and mean free paths at high fuel temps Resonant energy neutrons STILL get captured Off-Resonant energy neutrons now get captured This can be explained by effects of self-shielding within fuel pellet ELO 2.2

Doppler Broadening Summary
At low fuel temperatures Peaks are taller, but narrow Less overall resonance capture At high fuel temperatures Peaks are shorter, but broader More overall resonance capture ELO 2.2

Self Shielding At low fuel temperatures
Resonant energy neutron has high probability of absorption most likely in outer edge of fuel pellet Off-Resonant energy neutrons pass directly through pellet without being absorbed High self-shielding (of inner part of pellet) At high fuel temperatures Resonant energy neutrons have lower probability of absorption But still get captured Most likely towards middle of pellet Off-Resonant energy neutrons also get captured in pellet Less self-shielding Nuclear fuel (PWR) is ceramic pellets housed in a helium gas-filled, Zircaloy-clad, cylindrical fuel pin Neutrons are slowed down (thermalized) in surrounding moderator High-energy neutrons pass through fuel pellets and surrounding cladding into the moderator Moderator slows neutrons down into epithermal (intermediate) and thermal energy range Picture on next slide helps explain. ELO 2.2

Fuel Pellet Design - Self Shielding
Low Power An epithermal neutron that is not at resonance energy as it penetrates deeper into a pellet may appear as a resonance energy neutron Because temperature gradient is not as large as that found in a pellet located in a higher power region of the core, off-resonance energy neutron may pass completely through pellet and not be captured Figure: Fuel Pellet Shielded Areas ELO 2.2

Fuel Pellet Design - Self Shielding
High Power In a high power region of core, same neutron would have a higher probability of appearing as a resonance energy neutron upon entering the pellet and as it goes deeper into the pellet Result As fuel temperature increases, effective resonance capture area for epithermal neutrons also increases Only small fraction of these neutrons escape resonance capture in fuel pellet Figure: Fuel Pellet Shielded Areas ELO 2.2

Self Shielding Figure a: only resonance neutrons are captured, as shown by the 21 eV neutron Figure b: U-238 nuclei tend to capture both resonance and off- resonance neutrons Self-shielding is further explained by considering two uranium oxide fuel pellets, one at room temperature and another at operating reactor fuel temperature Figure: UO2 Fuel Pellet at Room and Operating Reactor Temperature ELO 2.2

Fuel Temperature Effects on Self Shielding
Two issues that must be considered to determine amount of self- shielding occurring: Physical size of the fuel pellet Design characteristics of the fuel pellet Nuclear reactor fuel pellets are uranium oxide ceramic pellets Poor conductors of heat Results in large temperature gradients from center to outer surface of pellet Major contributor to reduction in self-shielding as fuel temperature is increased These two effects determines overall effect of fuel temperature on resonance capture The next few slides show calculation of how neutron travel (average of three mean free paths) may or may not be captured in a fuel pellet of a given diameter based on fuel temperatures. None of this is testable it merely shows a relationship between macroscopic cross section and distance travelled by neutrons, as well as additional clarification on self-shielding. Feel free to go through the next few slides as quickly as desired. ELO 2.2

Mean Free Path Example Recall that mean free path (t) is defined as average distance that a neutron travels before being absorbed t= 1 Σa 𝑜𝑟 1 𝑁𝜎 𝑎 Where: t = mean free path (cm) Σa = macroscopic cross section (cm-1) N = atomic density (atoms/cm3) σa = microscopic cross-section for absorption (barns) Atomic density (N) is 2 x 1022 atoms/cm3 for the U-238 in a fuel pellet Assume every neutron is absorbed in three (3) mean free paths The physical size of the fuel pellets and the average distance that a neutron can travel into a pellet prior to resonance absorption occurring determines if a neutron will pass through the pellet without absorption. Even though mean free path and reactor period use the same Greek letter, mean free path is not currently used in any of the NRC exam bank questions. ELO 2.2

Mean Free Path Example Values used for this example (approximations):
Low temperatures Resonant Energy Neutrons 21 ev neutron σa = 5500 Barns Off-resonant energy neutrons 20 or 22 ev neutrons = σa 15 Barns High Temperatures Resonant energy neutrons 21 ev neutron σa = 500 Barns 20 or 22 ev neutrons = σa 250 Barns These microscopic cross sections are approximations but are representative of the change in resonance capture peaks of U-238 at high and low temps. ELO 2.2

Mean Free Path Example Resonant capture at low temp:
Three mean free paths 0.009 cm x 3 = cm < 1 cm Therefore, all neutrons at 21 eV entering fuel pellet will be absorbed Recall that 1 barn = cm2 ELO 2.2

Mean Free Path Example Off-Resonant capture at low temp:
Three mean free paths 3.33 cm x 3 = 10 cm, which is > 1 cm Therefore, none of the neutrons at 20 or 22 eV entering fuel pellet will be absorbed Recall that 1 barn = cm2 ELO 2.2

Mean Free Path Example Resonant capture at high temp:
Three mean free paths 0.1 cm x 3 = 0.3 cm, which is < 1 cm Therefore, all neutrons at 21 eV entering fuel pellet will STILL be absorbed Recall that 1 barn = cm2 ELO 2.2

Mean Free Path Example Off-Resonant capture at high temp:
Three mean free paths 0.2 cm x 3 = 0.6 cm, which is < 1 cm Therefore, all of the neutrons at 20 or 22 eV entering fuel pellet will be absorbed (at least most) Recall that 1 barn = cm2 ELO 2.2

Fuel Pellet Design Characteristics
This figure shows temperature gradients encountered for fuel pellets located in low and high power regions of the core. Comparing the two gradient curves for high and low temperature conditions, it can be seen that as center temperatures increase the delta temperature across the fuel pellet increases as well. For fuel pellets in high power regions of the core, the fuel centerline temperatures may be above 3000 °F, while temperatures near the fuel pellet surface are closer to 1000 °F. For fuel pellets in lower power regions of the core, the centerline temperature may be 1500 °F whereas the temperature at the surface of the pellet is closer to 700 °F. Figure: Fuel Pellet Temperature Profile ELO 2.2

Self Shielding Summary
More self-shielding at low fuel temps Resonant neutrons captured at edges of pellet Off-Resonant neutrons pass through pellet Less self-shielding at high fuel temps Resonant neutrons STILL captured, but towards center of pellet Off-Resonant neutrons NOW captured towards center of pellet Area under resonance peak curve same at low temps or high temps In Conclusion: More neutrons captured in resonance peaks at high fuel temps Due to less self-shielding ELO 2.2

Doppler Broadening/Self-Shielding
Knowledge Check If fuel temperature decreases by 50°F, the area under the resonance peak curve will ___________ and positive reactivity will be added to the core because ____________. decrease; fewer neutrons will be absorbed by U-238 overall decrease; fewer 6.7 eV neutrons will be absorbed by U-238 at the resonance energy remain the same; fewer neutrons will be absorbed by U overall remain the same; fewer 6.7 eV neutrons will be absorbed by U-238 at the resonance energy Correct answer is C. Correct answer is C. NRC Bank Question – P2751 Analysis: The area under the curve remains the same Essentially the same number of 6.7 ev neutrons get captured but less off-resonant energy neutrons get captured. Therefore, less “overall” neutrons get captured. Even though the area under the curve remains the same, there aren’t fewer 6.7 ev neutrons captured. When temperature lowers the microscopic cross section for 6.7 ev neutrons actually increases (peak gets taller) ELO 2.2

Fuel Temperature Coefficient Variations
ELO 2.3 – Describe how the magnitude of the fuel temperature coefficient varies with changes in the following parameters: moderator temperature, fuel temperature, and core age. Fuel temperature coefficient (FTC) values not constant throughout core life Various core parameters affect reactivity worth of FTC Analysis similar to MTC Only ONE change analyzed at a time Related KA - K1.07 Describe the effects of core age, fuel temperature, and moderator temperature on fuel temperature (doppler) coefficient ELO 2.3

Moderator Temperature Effects on FTC
Recall – FTC is how much negative reactivity is added when fuel temperature is raised one degree At lower moderator temperature (Case °F) Neutrons travel less (better moderation) When fuel temperature increased and peaks broaden Less resonance capture occurs At higher moderator temperature (Case °F) Neutrons travel farther (less moderation) More resonance capture occurs Summary – FTC more negative at HIGH moderator temperatures Sanity Check: This one can confuse students because we are trained that heat transfer is always from HOT to COLD. Seems strange to raise fuel temperature from one temp to another temp with varying moderator temperatures. However, this could happen if steam demand is changed. At low temperatures, neutron slowing down time and length very short Resonance capture is reduced Therefore, changes in resonant absorption peaks (Doppler) will cause a relatively smaller effect on the Doppler coefficient When the moderator is hot (less dense) or contains voids, slowing down length and time for neutrons increases Changes in resonance absorption peaks (Doppler) now more significant since neutrons are spending longer periods of time in resonance energy range Doppler coefficient (FTC) is more negative at high moderator temperatures and is most negative at high void fractions For this example, assume fuel temperature is raised from 1500°F – 1501°F with RCS temperature at 557°F Case 1 Case 2 Tfuel °F Tfuel °F Tmod - 557°F Tmod - 587°F ELO 2.3

Fuel Temperature Effects on FTC
At low fuel temperatures (raising from 1000°F – 1001°F) Resonance capture peaks are narrow and tall Small increase in fuel temperature causes Significant increase in vibrational energy of U-238 MANY neutrons captured in resonance peaks Net result - FTC is relatively high At high fuel temperatures (raising from 1500°F – 1501°F) Resonance capture peaks are broad and short Smaller increase in vibrational energy of U-238 Not as many neutrons captured in resonance peaks Net result - FTC is relatively low Summary – FTC is more negative at low fuel temperatures This parameter you need to look at the “fractional change in fuel temperature” to answer these questions. For example, raising fuel temperature from 1000°F to 1001°F is a bigger percentage change than raising it from 1500°F to 1501°F . Even though raising fuel temperatures ALWAYS results in negative reactivity being added, each degree at low fuel temps (low power) adds more negative reactivity than each degree at high fuel temps (high power). One way the NRC answers this is that at high temps there is “minimal change to Doppler broadening” (peaks are about as broad as they can get). Visual on next slide. ELO 2.3

Fuel Temperature Effects on FTC
Slope of curve greater at low fuel temperatures This figure illustrates that a 1 °F change from 1000 F to 1001 F results in a larger magnitude of change for the Doppler coefficient (D) than a 1 F change from 3000 F to 3001 F. This is a result of the magnitude of the Doppler broadening change of the uranium-238 (and plutonium-240) target nuclei being greater at lower fuel temperatures rather than higher. Note that FTC is always negative. Its magnitude is simply smaller at higher fuel temperatures. Figure: Change in Magnitude of Doppler Coefficient versus Fuel Temperature ELO 2.3

Core Age Affects on FTC At BOL
U-238 and U-235 isotopes present in core These fuels cause a significant amount of resonance capture U-238 is the greater of these two isotopes Recall for U-235, 86% of time fissions, 14% of time captures At EOL Approximately same U-238 U-235 reduced to 60 percent Pu-239 and Pu-240 now present Pu-239, 73% of time fissions, 27% of time captures For this example, assume fuel temperature is raised from 1500°F – 1501°F at BOL and EOL Case 1 Case 2 Tfuel °F Tfuel °F BOL EOL The only difference are the isotopes available to capture neutrons. At BOL there is only U At EOL there is U-238 AND Pu Therefore, FTC is more negative at EOL. ELO 2.3

Plutonium Production 238 92 𝑈+ 1 0 𝑛 𝜎 𝑦 =277𝑏 239 92 𝑈
𝑈+ 1 0 𝑛 𝜎 𝑦 =277𝑏 𝑈 𝑈 𝛽 − ,𝛾 𝑁𝑝 𝑡 1/2 =23.5 𝑚 𝑁𝑝 𝛽 − ,𝛾 𝑃𝑢 𝑡 1/2 =2.355 𝑑 𝑃𝑢+ 1 0 𝑛 𝜎 𝑦 =270𝑏 𝑃𝑢 Plutomium-239 produces plutonium-240 approximately 27% of the time - neutron capture. 73% of the time plutonium-239 results in fission. Recall that Pu-239 is considered fissile (or fertile) so it likes thermal neutrons. ELO 2.3

Pu-240 Absorption Cross Section
This figure shows the total cross-section for plutonium-240. The capture cross-section represents the largest component of the total cross-section for plutonium-240. Notice the high resonance peak of 105 barns (for about 1 – 2 ev neutrons), Figure: Total Cross-section for Plutonium-240 ELO 2.3

Core Age Affects on FTC Due to buildup of Pu-240 over core life:
FTC more negative at EOL Pu-240 has high capture cross-section for 1 eV neutrons (approximately 1  105 barns) ≈ Same amount of U-238 in core at EOL Also, at EOL better heat transfer Clad creep and Pellet swell Results in a slightly lower fuel temperature FTC more negative at low fuel temperatures Concept explained in greater detail in Thermodynamics ELO 2.3

Core Age Affects on FTC Typical values for FTC over core life:
-1 x 10-5 Δk/k/°F at BOL -1.5 x 10-5 Δk/k/°F at EOL Keep in mind that the “typical values” don’t exactly line up with the values shown on the curve. Main point is that FTC becomes more negative over core life due to the buildup of Pu-240 as well as a slightly lower fuel temperature at EOC. Also note that the average fuel temperature will rarely exceed 2000oF. Figure: Value of Doppler Coefficient vs. Temperature over Core Life ELO 2.3

Fuel Temperature Coefficient Variations
Knowledge Check A nuclear power plant is operating at steady-state 70 percent power. Which one of the following will result in a less negative fuel temperature coefficient? (Consider only the direct effect of the change in each listed parameter.) Increase in Pu-240 inventory in the core. Increase in moderator temperature. Increase in fuel temperature. Increase in coolant voids. Correct answer is C. Correct answer is C. NRC Bank Question – P1951 NOTE: When reading choices relating to coefficients, you MUST understand what is being asked. The STEM of the question is asking when is FTC “less” negative. “A” should be read, “Is FTC less negative with more Pu-240 in the core?” NO! “B” should be read, “Is FTC less negative at high moderator temperatures?” NO! “C” should be read, Is FTC less negative at high fuel temperatures?” YES! “D” should be read, Is FTC less negative with high voiding?” NO! Analysis: A. WRONG. A higher concentration of Pu-240 in the core results in additional isotopes that have large resonant peaks, resulting in more negative reactivity inserted for a rise in fuel temperature, thus, a more negative fuel temperature coefficient. B. WRONG. FTC is more negative at higher moderator temperatures because neutrons are traveling farther. C. CORRECT. At higher fuel temperatures (large Doppler broadening), the fuel temperature coefficient (negative reactivity inserted per degree fuel temperature rise) will become less negative. This is because the amount of Doppler broadening per degree change in fuel temperature lowers; at lower temperatures a one-degree rise in fuel temperatures results in more vibration/Doppler broadening of the fuel compared to higher temperatures. D. WRONG. This is essentially the same as an increase in moderator temperature. ELO 2.3

Fuel Temperature Coefficient Changes
Knowledge Check Which one of the following pairs of isotopes is responsible for most of the negative reactivity associated with a fuel temperature increase near the end of core life? U-235 and Pu-239 U-235 and Pu-240 U-238 and Pu-240 U-238 and Pu-239 Correct answer is C. Correct answer is C. NRC Bank Question – P1851 Analysis: As core life increases, the FTC becomes more negative at EOL due to production of Pu-240 , which has more resonance peaks that broaden at higher temperatures (near 1ev). While these isotopes raise the probability of resonant absorption, throughout core life U-238 is the most significant contributor to resonant capture due to relative abundance compared to other isotopes. One thumbrule for determining which isotopes are more prone to resonant capture: EVEN # isotopes are more likely to resonant capture neutrons; ODD # isotopes are more likely to fission. ELO 2.3

Power Coefficient ELO 2.4 – Describe the components of the power coefficient of reactivity and the magnitude of their overall effect over core life. Power coefficient of reactivity (αPower) combines MTC, FTC, and VC In some plants, void coefficient may be disregarded Reactivity change per unit change in power pcm/% power Unless stated otherwise assumes temperature starts and ends on program Related KA – K1.08 Describe the components of power coefficient Related coefficients combined into single coefficient called power reactivity coefficient Without the power coefficient reactivity adjustments would require considering MTC and FTC separately. We don’t have fuel temperature monitoring! ELO 2.4

Power Coefficient Equation for the power coefficient of reactivity (αPower) is similar to the equations for other reactivity coefficients: 𝛼 𝑃𝑜𝑤𝑒𝑟 = ∆𝜌 ∆% 𝑃𝑜𝑤𝑒𝑟 Where: αPower = power coefficient of reactivity (pcm/% power) Δρ = change in reactivity from change in power (pcm) Δ% Power = change in reactor power (% power) MTC is slow acting because the fuel must first heat up, then transfer heat to the moderator/coolant FTC quickest acting of all of reactivity coefficients because an increase in power results in an immediate change in fuel temperature In reactor design, essential that both MTC and FTC be negative If power is increased due to a positive reactivity insertion, resultant increase in fuel temperature and moderator temperature adds negative reactivity to turn or limit the power increase ELO 2.4

MTC, FTC, and PC Values MTC varies from:
+1 pcm/°F at BOL (if +MTC allowed) -26 pcm/°F at EOL FTC changes from: -1 pcm/°F at BOL -1.5 pcm/°F at EOL Estimated changes in Power Coefficient (PC): -15 pcm/% power at BOL -22 pcm/% power at EOL If MTC and FTC are more negative at EOL, then PC is also MTC values will vary between plants. (might not be allowed to be +, might be as negative as -35). Remind students of “Various GFE Values.pdf” file that has some of these values on them. Also recall that SDM decreases over core life because on a trip, the Power Coefficient adds more positive reactivity. ELO 2.4

Power Coefficient Knowledge Check
Which one of the following groups contains parameters that, if varied, will each have a direct effect on the power coefficient? Control rod position, reactor power, moderator void fraction Moderator temperature, reactor coolant system pressure, xenon-135 concentration Fuel temperature, xenon-135 concentration, control rod position Moderator void fraction, fuel temperature, moderator temperature Correct answer is D. Correct answer is D. NRC Bank Question – P253 ELO 2.4

Power Defect Affect on Reactor Power Operations
ELO 2.5 – Explain how the power defect affects the reactivity balance on reactor power operations. Power Defect = Power Coefficient x change in power -pcm/% power x (+/-) % power change = (+/-) pcm Because of power defect When power is raised, negative reactivity is added, therefore, Rods must be withdrawn, and/or Boron concentration reduced The power coefficient and defect add to the inherent safety features of a commercial PWR. ELO 2.5

Power Defect at EOL At EOL conditions:
Moderator temperature defect = -26 PCM/°F X 30°F = -780 PCM Fuel temperature defect = -1.5 PCM/°F X 1,000°F = -1,500 PCM Power defect = pcm/% power X 100% = -2,280 PCM These values provide an idea of how much reactivity must be compensated for during power changes 𝑀𝑜𝑑 𝑇𝑒𝑚𝑝 𝐷𝑒𝑓𝑒𝑐𝑡 + 𝐹𝑢𝑒𝑙 𝑇𝑒𝑚𝑝 𝐷𝑒𝑓𝑒𝑐𝑡 = 𝑃𝑜𝑤𝑒𝑟 𝐷𝑒𝑓𝑒𝑐𝑡 Note: The moderator temperature defect assumes a 30°F increase in RCS average temperature from 0 to 100% power; Fuel temperature rise of 1000°F is assumed ELO 2.5

Power Defect Calculation Example
Given the requirement to increase power from 20% to 80% and the following reactivity values describe the boron concentration change needed. Control rods are at 150 steps and are to be withdrawn to 190 steps. Power Coefficient = -22 pcm/%. Control Rod worth = 5 PCM/step Boron worth = 7 pcm/ppm Example Given the requirement to increase power from 20 percent to 80 percent and the following reactivity values, describe the boron concentration change needed. Control rods are at 150 steps and are to be withdrawn to 190 steps. Power coefficient = -22 pcm/percent Control rod worth = 5 PCM/step Positive worth on withdrawal, negative worth on insertion Boron worth = -7 pcm/ppm ELO 2.5

Power Defect on Reactor Power Operations
Example: Given the requirement to increase power from 20% to 80% and the following reactivity values describe the boron concentration change needed. Control rods are at 150 steps and are to be withdrawn to 190 steps. Power Coefficient = -22 pcm/%. Control Rod worth = 5 PCM/step Boron worth = 7 pcm/ppm Solution Increase power 20% to 80% = −22 𝑝𝑐𝑚 % 𝑝𝑜𝑤𝑒𝑟 ×60% 𝑝𝑜𝑤𝑒𝑟 =−1,320 𝑝𝑐𝑚; therefore, pcm must be added Rod withdrawal 150 to 190 steps = 5 𝑃𝐶𝑀 𝑠𝑡𝑒𝑝 ×40 𝑠𝑡𝑒𝑝𝑠=200 𝑝𝑐𝑚 Remaining reactivity from boron = 1,320 𝑝𝑐𝑚−200 𝑝𝑐𝑚 =+1,120 𝑝𝑐𝑚 Boron concentration change for 1,120 pcm = 1,120 𝑝𝑐𝑚 −7 𝑝𝑐𝑚 𝑝𝑝𝑚 =−160 𝑝𝑝𝑚 Boron concentration change needed is to dilute 160 ppm ELO 2.5

Power Coefficient on Reactor Operations
Knowledge Check A nuclear power plant is operating at steady-state 100 percent power. Given the following initial parameters, select the final reactor coolant boron concentration required to decrease average coolant temperature by 6°F. (Assume no change in control rod position or reactor/turbine power.) Reactor coolant boron concentration = 500 ppm Moderator temperature coefficient = %ΔK/K/°F Differential boron worth = %ΔK/K/ppm 509 ppm 504 ppm 496 ppm 491 ppm Correct answer is A. Correct answer is A. NRC Bank Question – P1553 Perform calculation on the board. Important concept is: The STEM asks how much boron is required to “decrease average coolant temperature…”. To lower RCS temp you raise boron concentration. Therefore, Choices “C” and “D” can be eliminated. Short answer - -6 (decrease in temp) x -12 (pcm/degree)= +72 pcm. Boron needs to be -72 pcm. -72/-8 (pcm/ppm) = +9 ppm = 509 ppm. ELO 2.5

Boron Reactivity Control
TLO 3 – Describe boron worth and how boron is utilized to control the reactor from beginning to end of core life. 3.1 Describe differential and integral boron. 3.2 Describe how boron reactivity worth changes with boron concentration and moderator temperature. 3.3 Explain the change in reactivity addition rate resulting from changing boron concentration over core life. NOTE – the term “differential” is the same as “coefficient” and “integral” is the same as “defect”. TLO 3

Boron Reactivity Control
ELO 3.1 – Describe differential and integral boron worth and the factors that affect it. Boron affects thermal utilization factor As you change boron concentration you change “competition” High absorption cross section for thermal neutrons Considered a 1/v absorber Linear increase in cross section as neutron slows down Adds to macroscopic cross section of moderator High concentration at BOL Offsets Kexcess Related KA K1.22 Explain how boron concentration affects core life ; K1.04 Describe how and why boron concentration changes over core life This term originates from the movement of control rods to control reactivity. Inserting and withdrawing (shimming) a reactor’s control rods varies the reactivity present in the core. Similarly, adjusting the concentration of boron in the coolant (chemical shim) affects the amount of reactivity present in the reactor. Control rods are discussed in greater detail later. Related to 1/v absorber: There is a goofy question in the NRC exambank (P1350) that talks about the effect on boron worth with boron in the core. Since boron likes thermal neutrons, if there are a lot of boron in the core it tends to push the average flux in the core towards the epithermal region, where boron’s absorption cross section is lower. Not sure what the knowledge benefit this is, but it is in the bank! ELO 3.1

Differential Boron Worth
The term “differential” similar to the term “coefficient” Unit change in reactivity/change in boron concentration pcm/ppm Unit changes vary in worth over varying plant conditions Boron concentration Moderator temperature ELO 3.1

Integral Boron Worth The term “integral” similar to “defect”
Total change in reactivity pcm/ppm x Dppm = pcm Used in various calculations for power changes Compensates with control rods for Power Defect during power changes ELO 3.1

Boron Worth Changes with Boron Concentration
Below is a macroscopic cross-section for absorption versus boron concentration Cross-section varies linearly with boric acid concentration CB (ppm) Σa Boron (cm-1) Σa B and H2O (cm-1) 0.022 500 0.021 0.043 1,000 0.042 0.064 1,500 0.063 0.085 2,000 0.084 0.106 2,500 0.105 0.127 The variation of the macroscopic cross-section for absorption (Σa) with boron concentration is shown in this table. It can be seen, as expected, that macroscopic absorption cross-sections of boron dissolved in water increase with concentration (ppm). The cross-section also varies linearly with boric acid concentration. Recall, at high boron concentrations, the moderator is a better “absorber” than “scatterer” (+MTC) ELO 3.1

Boron Concentration Changes Over Core Life
Boric acid added to RCS to help control kexcess At BOL, IBW about -20,000 pcm cold shutdown and borated to about 2,000 ppm DBW at BOL (cold) -10 to -11 pcm/ppm DBW at BOL (hot) -6 to -7 pcm/ppm DBW at EOL (hot) -8 pcm/ppm Boric acid is added to the reactor coolant system to help control excess reactivity (kexcess) from fuel loading and perform slow reactivity changes needed for control rod positioning and fission product poison compensation. A couple of things this graph shows related to factors affecting DBW (explained later): 1. DBW greater when cold (denser water) 2. DBW greater at EOL (less competition with itself) Figure: Differential Boron Worth ELO 3.1

Boron Concentration Changes Over Core Life
With a value of -10 pcm/ppm, negative reactivity from 2,000 ppm boron is: ∆𝜌 ∆ 𝐶 𝐵 = −10 𝑝𝑐𝑚 𝑝𝑝𝑚 ∆𝜌 𝑝𝑐𝑚 = −10 𝑝𝑐𝑚 𝑝𝑝𝑚 ×2,000 𝑝𝑝𝑚 ∆𝜌 𝑝𝑐𝑚 =−20,000 𝑝𝑐𝑚 Based on Kexcess discussion in earlier chapter, at BOL Kexcess about – pcm. If boron compensates for pcm then burnable poisons must account for the rest. Once at power and Xe and Sm build in, boron concentration can be reduced. ELO 3.1

Boron Reactivity Worth Variations
ELO 3.2 – Describe how boron reactivity worth changes with boron concentration and moderator temperature. Boron reactivity worth (pcm/ppm or Δk/k/ppm) is affected by changes in boron concentration and moderator temperature With boron concentration approaching 2,000 ppm at BOL and less than 50 ppm EOL, concentration differences are large Changes in boron concentration similar to changes in core life Related KAs K1.09 Compare boron reactivity worth vs. boron concentration ; K1.10 Compare boron reactivity worth vs. moderator temperature Changing boron concentration at high concentration versus low concentration is basically the same thing as BOL versus EOL. Refer to bottom right of 11 x 17 paper brain for synopsis of various worth’s! When asnswering questions relating to DBW, remember the definition – DBW = How much reactivity is added by raising boron concentration one ppm? Since DBW is a negative value, it adds negative reactivity when concentration is raised. ELO 3.2

Boron Worth vs Boron Concentration
At BOL, high boron concentration Integral Boron Worth (IBW) greater at BOL At EOL, lower boron concentration For a given temperature: Higher % change in boron concentration from 100–101 ppm than 1000–1001 ppm DBW more negative at EOL Less competition with itself Notice slope of curve When discussing differential or coefficient, keep in mind that it is “how much reactivity is added per unit change in parameter”. When boron concentration is added at 100 ppm (to 101 ppm), the additional boron has less boron to compete with (less competition with itself), therefore, that unit change is worth more negative reactivity. Additionally, as boron concentration becomes greater with so many boron atoms in moderator, self-shielding between boron atoms occurs Decreases probability that an individual boron atom will absorb a given neutron Resulting in a decrease in differential boron worth for higher boron concentrations Differential boron worth figure also shows that differential boron worth at 578F is a lower magnitude (less negative) than at 78F Fewer boron atoms due to lower moderator density at higher temperature Figure: Differential Boron Worth ELO 3.2

Boron Worth vs Moderator Temperature
As temperature goes up, lower density (atoms/cm3), boron worth GOES DOWN (less negative)! As temperature goes down, higher density (atoms/cm3), boron worth GOES UP (more negative)! DBW more negative at lower temperatures! It is important to not confuse changes in differential boron worth (DBW) at different temperatures based on the number of atoms added. KEY POINT: DBW is worth more at lower temperatures because when boron concentration is raised one ppm, more atoms per cubic centimeter are added to the core (RCS is denser). At higher temperatures, reactor core contains a smaller mass of water (not volume) due to expansion of water Smaller water mass means a smaller mass of boron in core for a given concentration of boron (ppm) Lower boron density in core results in lower boron differential worth Figure: Reactivity Worth of Boron versus Moderator Temperature ELO 3.2

Boron Reactivity Worth Variations
Knowledge Check Differential boron worth (Δk/k/ppm) will become __________ negative as moderator temperature increases because, at higher moderator temperatures, a 1 ppm increase in reactor coolant boron concentration will add __________ boron atoms to the core. less; fewer more; fewer more; more less; more Correct answer is A. Correct answer is A. NRC Bank Question P1152 – modified order Analysis: If you raise boron concentration one ppm at an RCS temperature of 580°F, you are adding fewer atoms/cm3 in the core than if you raised it one ppm at an RCS temperature of 540°F. This makes the macroscopic cross section smaller at higher RCS temperatures. Therefore, for that one ppm change at higher RCS temperatures, boron has a smaller effect on reactivity (less negative). ELO 3.2

Boron Reactivity Addition Rates
ELO 3.3 – Explain the change in reactivity addition rate resulting from changing boron concentration over core life. This section describes how soluble boron concentration changes over core life and why Amount of water required for dilution changes over core life More water (about 10 times) used to dilute boron at EOL than BOL Same water used to borate at BOL and EOL (assuming concentration in the tank is constant) Time to make change reactivity changes over core life Faster at BOL (more boron in core) Related KA - K1.11 Explain the change in reactivity addition rate due to boration/dilution over core life Remind students that even though the reactor will respond faster at EOL (smaller Beta-Bar-Effective), you can add the reactivity faster at BOL (higher boron concentration) ELO 3.3

Purpose for Boron Reactivity Control
Boron dilution used with control rod withdrawal Offset negative reactivity of power defect during power ascension Rods usually withdrawn by 50% power Limit axial flux distribution problems Boron dilution then used: Continued power ascension to 100% Compensate for buildup of fission product poisons Compensate for fuel depletion Some plants do a boron “dilution to criticality” after a refueling outage. Soluble boron controls effects of slower reactivity changes Reduces overall control rod reactivity requirements in a PWR Allows for increased fuel loading and optimum positioning of control rods Combination of multi-region fuel loading and control rod programming reduces peak to average power density Using boron with control rods fully withdrawn (at 100% power) helps produce design radial and axial power distribution within the core “Flattening” power distribution in core minimizes possibility of excessively high heat flux in any one fuel rod ELO 3.3

Mechanism for Reactivity Control with Boron
Fuel depletes, adding negative reactivity Temperature decreases, adding positive reactivity Brings keff back to 1.0 Boron concentration is diluted adding positive reactivity Temperature increases back to program adding negative reactivity Net steady-state effect: No change in Rx power (steam demand) RCS temp stays on program over core life Soluble boron is uniformly mixed and circulated through the reactor via the moderator/coolant. Initial boron concentration present in moderator/coolant based on amount of excess reactivity (ρexcess) at BOL to permit extended operations at 100 percent Increasing boron concentration increases macroscopic cross-section for absorption of moderator/coolant Thermal utilization factor (f) decreases, reducing keff As fuel depleted during operation, core reactivity decreases Removing boron by dilution with pure water reduces boron concentration to add positive reactivity (f is increased) Rebalancing reactivity to maintain keff at 1.0 ELO 3.3

Boron Concentration Over Core Life
Initial drop due to buildup of fission product poisons ≈ 2 days for Xe, 2-3 weeks for Sm (clean core) Flat section Burnable poisons deplete at same rate as fuel depletion Might need to borate Remainder of graph Due to fuel depletion and buildup of “other” fission product poisons In this example, BOL boron concentration is about 1,200 ppm Remind the students of the Kexcess graph in – Neutron Life Cycle where it showed an increase in Kexcess. The “flat line section” could actually have an increase in boron concentration if the burnable poisons deplete faster than the fuel depletes. One reminder when answering bank questions: if asked “why do we dilute” (add positive reactivity) – look for answers that add negative reactivity. For example, why do we dilute (+ reactivity) in the last section of graph? The correct answer would be a) fuel depletion (- reactivity), and b) buildup of fission product poisons (- reactivity). If a choice of “burnable poisons deplete” is offered, that would be incorrect because that adds positive reactivity. Figure: Critical Boron Concentration over Core Life ELO 3.3

Boron Concentration - PPM
Chemical shim (boron) is a solution of boric acid (H3BO3) and water Concentration measured in parts per million of boron by weight 𝐶 𝐵 = 𝑔𝑟𝑎𝑚𝑠 𝑜𝑓 𝑏𝑜𝑟𝑜𝑛 𝑔𝑟𝑎𝑚𝑠 𝑜𝑓 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 When 1,000 ppm of boron present in solution, this ratio is: 1,000 𝑝𝑝𝑚= 𝑔𝑟𝑎𝑚𝑠 𝑜𝑓 𝑏𝑜𝑟𝑜𝑛 1.0 𝑔𝑟𝑎𝑚𝑠 𝑜𝑓 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 Changing concentration of soluble boron is a slow process Maximum rate of change in reactivity approximately 3 pcm/second Because of this, normally used for compensating for slowly changing reactivity parameters Fuel depletion Changes in fission product poison concentration However, boron addition and dilution also used during transient conditions to maintain control rod position/axial flux in desired range In such cases, boron changes must be anticipated to allow lead time for their desired reactivity effects to take place An example of this is when reactor is required to be shutdown from 100 percent In such cases, boron addition is required in addition to inserting control rods to counter positive reactivity added from power defect In this case, control rods do not provide sufficient negative reactivity since they are still required to be above their insertion limits for shutdown margin requirements ELO 3.3

Boration/Dilution over Core Life
Boration over core life Concentration of boron used to borate is same from BOL to EOL Water to borate over core life also doesn’t change Dilution over core life Concentration reduced by factor of 10 (BOL to EOL) Water to dilute increases by factor of 10 (BOL to EOL) Reactivity added by dilution Faster at BOL Higher concentration Over time [B] slowly decreases (no operator action) Doing its job of capturing thermal neutrons Plant Curve books include correction factors for this Concentration of boric acid used for boron addition is constant over core life Therefore, a given amount of boric acid will produce same ppm change in boron concentration at any time in core life Dilutions are a different matter From BOL to EOL, boron concentration reduced over a factor of 10 At core EOL, 10 times as much water must be removed to have the same ppm decrease in boron concentration as at BOL Every gallon of borated water removed through dilution will carry with it less boron at core EOL than at core BOL Therefore, for positive reactivity additions via dilution, much more water is required, and more time (gallons per minute) is necessary to effect same positive reactivity addition Over time boron concentration naturally decreases by absorbing thermal neutrons. Correction factors are normally included in Plant Curve books when diluting over core life because of this phenomenon. This concept is tested in – Fuel Depletion/Burnable Poisons. ELO 3.3

Boron Reactivity Addition Rates
Knowledge Check The amount of pure water required to decrease the reactor coolant boron concentration by 20 ppm at the beginning of core life (1000 ppm) is approximately ______________ the amount of pure water required to decrease reactor coolant boron concentration by 20 ppm at the end of core life (100 ppm). one-tenth the same as 10 times 100 times Correct answer is A. Correct answer is A. NRC Bank Question – P1050 – Modified Since boron concentration essentially decreases by a factor of 10 over core life, the amount of water required to make a given change increases by a factor of 10 over core life. This is an example of how a provided bank question can be modified to change the answer without changing any of the choices. Based on wording of actual bank question, Choice “C” was correct. By changing the comparison in this modified question, Choice “A” is correct. BE CAREFUL WHEN READING BANK QUESTIONS! ELO 3.3

Reactivity Coefficients Summary
MTC MORE negative: At EOL (lower boron concentrations) Less boron in core At high power/temperature Greater change in density FTC MORE negative: At EOL Building up Pu-240 and lower fuel temperature At low fuel temperatures Greater change in Doppler broadening (and temp) At high RCS temperatures Neutrons are traveling farther Resonance Absorption Peaks At low fuel temps – taller and narrow At high fuel temps – shorter and broader Summary

Reactivity Coefficients Summary
Power Coefficient more negative: EOL MTC and FTC are more negative at EOL Power Defect (PD) components: MTD, FTD, VD FTD biggest part of these DBW more negative: Lo RCS temperatures Higher density (more atoms/cm3 added) Less competition with itself (greater % change in boron) IBW more negative: BOL Higher boron concentration MTD – Moderator Temperature Defect FTD – Fuel Temperature Defect VD – Void Defect Summary

KA to ELO Tie KA # KA Statement RO SRO ELO K1.01 Define moderator temperature coefficient of reactivity. 3.1 3.2 1.2 K1.02 Define fuel temperature coefficient of reactivity. 3.0 2.1 K1.03 Describe the effect on the magnitude of the temperature coefficient of reactivity from changes in moderator temperature and core age. 2.9 1.3 K1.04 Explain resonance absorption. 2.4 2.2 K1.05 Explain doppler broadening and self-shielding. 2.3 K1.06 Describe time effects of core age, moderator temperature, and boron concentration on moderator temperature coefficients. K1.07 Describe the effects of core age, fuel temperature, and moderator temperature on fuel temperature (doppler) coefficient. K1.08 Describe the components of power coefficient. K1.09 Compare boron reactivity worth vs. boron concentration. 2.8 K1.10 Compare boron reactivity worth vs. moderator temperature. K1.11 Explain the change in reactivity addition rate due to boration/dilution over core life. 3.3 K1.12 Explain differences between reactivity coefficients and reactivity defects. 2.7 1.1 K1.13 Explain and describe the effect of power defect and doppler defect on reactivity. 2.5 NOTE: ELO-3.1 not specifically covered by a KA in this chapter. However, this knowledge is tied to KA’s in future Rx Theory chapters.

Operator Generic Fundamentals Reactor Theory - Reactivity Coefficients

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Operator Generic Fundamentals Reactor Theory - Reactivity Coefficients

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